JP-A-2006-274963 discloses an invention for changing the amplitude of air-fuel ratio fluctuation in perturbation control corresponding to fluctuation in exhaust gas volume in an apparatus for performing perturbation control where an exhaust air-fuel ratio is alternately changed to a rich side and a lean side with respect to a stoichiometric air-fuel ratio to quickly warm up a three-way catalyst at the time of cold starting of an engine.    Patent Document 1: JP-A-2006-274963    Patent Document 2: JP-A-2006-9674